MathDB

2003 AMC 8

Part of AMC 8

Subcontests

(25)
17
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2003 AMC 8 Problem 17- Sibblings of Jim

The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings? ChildEye ColorHair ColorBenjaminBlueBlackJimBrownBlondeNadeenBrownBlackAustinBlueBlondeTevynBlueBlackSueBlueBlonde \begin{array}{c|c|c}\text{Child}&\text{Eye Color}&\text{Hair Color}\\ \hline \text{Benjamin}& \text{Blue} & \text{Black} \\ \hline \text{Jim} & \text{Brown} & \text{Blonde} \\ \hline \text{Nadeen} & \text{Brown} & \text{Black}\\ \hline \text{Austin}& \text{Blue} & \text{Blonde}\\ \hline \text{Tevyn} & \text{Blue} & \text{Black} \\ \hline \text{Sue} & \text{Blue} & \text{Blonde} \\ \hline \end{array}
<spanclass=′latex−bold′>(A)</span> Nadeen and Austin<spanclass=′latex−bold′>(B)</span> Benjamin and Sue<spanclass=′latex−bold′>(C)</span> Benjamin and Austin<span class='latex-bold'>(A)</span>\ \text{Nadeen and Austin} \qquad <span class='latex-bold'>(B)</span>\ \text{Benjamin and Sue}\qquad <span class='latex-bold'>(C)</span>\ \text{Benjamin and Austin}\qquad <spanclass=′latex−bold′>(D)</span> Nadeen and Tevyn<spanclass=′latex−bold′>(E)</span> Austin and Sue<span class='latex-bold'>(D)</span>\ \text{Nadeen and Tevyn} \qquad <span class='latex-bold'>(E)</span>\ \text{Austin and Sue}
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2003 AMC 8 Problem 14-System of equations

In this addition problem, each letter stands for a different digit. \setlength{\tabcolsep}{0.5mm}\begin{array}{cccc}&T & W & O\\ \plus{} &T & W & O\\ \hline F& O & U & R\end{array} If T = 7 and the letter O represents an even number, what is the only possible value for W?
<spanclass=′latex−bold′>(A)</span> 0<spanclass=′latex−bold′>(B)</span> 1<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 3<spanclass=′latex−bold′>(E)</span> 4<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ 2\qquad <span class='latex-bold'>(D)</span>\ 3\qquad <span class='latex-bold'>(E)</span>\ 4
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AMC 8 2003 Problem 23 - Cat and Mouse

In the pattern below, the cat (denoted as a large circle in the figures below) moves clockwise through the four squares and the mouse (denoted as a dot in the figures below) moves counterclockwise through the eight exterior segments of the four squares.
[asy]defaultpen(linewidth(0.8)); size(350); path p=unitsquare; int i; for(i=0; i<5; i=i+1) { draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); } path cat=Circle((0.5,0.5), 0.3); draw(shift(0,1)*cat^^shift(4,1)*cat^^shift(7,0)*cat^^shift(9,0)*cat^^shift(12,1)*cat); dot((1.5,0)^^(5,0.5)^^(8,1.5)^^(10.5,2)^^(12.5,2));
label("1", (1,2), N); label("2", (4,2), N); label("3", (7,2), N); label("4", (10,2), N); label("5", (13,2), N); [/asy]
If the pattern is continued, where would the cat and mouse be after the 247th move?
<spanclass=′latex−bold′>(A)</span><span class='latex-bold'>(A)</span> [asy]defaultpen(linewidth(0.8)); size(60); path p=unitsquare; int i=0; draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); path cat=Circle((0.5,0.5), 0.3); draw(shift(1,0)*cat); dot((0,0.5)); [/asy]
<spanclass=′latex−bold′>(B)</span><span class='latex-bold'>(B)</span> [asy]defaultpen(linewidth(0.8)); size(60); path p=unitsquare; int i=0; draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); path cat=Circle((0.5,0.5), 0.3); draw(shift(1,1)*cat); dot((0,0.5)); [/asy]
<spanclass=′latex−bold′>(C)</span><span class='latex-bold'>(C)</span> [asy]defaultpen(linewidth(0.8)); size(60); path p=unitsquare; int i=0; draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); path cat=Circle((0.5,0.5), 0.3); draw(shift(1,0)*cat); dot((0,1.5)); [/asy]
<spanclass=′latex−bold′>(D)</span><span class='latex-bold'>(D)</span> [asy]defaultpen(linewidth(0.8)); size(60); path p=unitsquare; int i=0; draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); path cat=Circle((0.5,0.5), 0.3); draw(shift(0,0)*cat); dot((0,1.5)); [/asy]
<spanclass=′latex−bold′>(E)</span><span class='latex-bold'>(E)</span> [asy]defaultpen(linewidth(0.8)); size(60); path p=unitsquare; int i=0; draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); path cat=Circle((0.5,0.5), 0.3); draw(shift(0,1)*cat); dot((1.5,0)); [/asy]
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2003 AMC 8 Problem 3-Percentage of burger

A burger at Ricky C's weighs 120 grams, of which 30 grams are filler. What percent of the burger is not filler?
<spanclass=′latex−bold′>(A)</span> 60%<spanclass=′latex−bold′>(B)</span> 65%<spanclass=′latex−bold′>(C)</span> 70%<spanclass=′latex−bold′>(D)</span> 75%<spanclass=′latex−bold′>(E)</span> 90%<span class='latex-bold'>(A)</span>\ 60\%\qquad <span class='latex-bold'>(B)</span>\ 65\% \qquad <span class='latex-bold'>(C)</span>\ 70\%\qquad <span class='latex-bold'>(D)</span>\ 75\% \qquad <span class='latex-bold'>(E)</span>\ 90\%
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AMC 8 2003 Problem 6 - Squares on Triangle Sides

Given the areas of the three squares in the figure, what is the area of the interior triangle?
[asy] real r=22.61986495; pair A=origin, B=(12,0), C=(12,5); draw(A--B--C--cycle); markscalefactor=0.1; draw(rightanglemark(C, B, A)); draw(scale(12)*shift(0,-1)*unitsquare); draw(scale(5)*shift(12/5,0)*unitsquare); draw(scale(13)*rotate(r)*unitsquare); pair P=shift(0,-1)*(13/sqrt(2) * dir(r+45)), Q=(14.5,1.2), R=(6, -7); label("169", P, N); label("25", Q, N); label("144", R, N); [/asy]
<spanclass=′latex−bold′>(A)</span> 13<spanclass=′latex−bold′>(B)</span> 30<spanclass=′latex−bold′>(C)</span> 60<spanclass=′latex−bold′>(D)</span> 300<spanclass=′latex−bold′>(E)</span> 1800 <span class='latex-bold'>(A)</span>\ 13\qquad<span class='latex-bold'>(B)</span>\ 30\qquad<span class='latex-bold'>(C)</span>\ 60\qquad<span class='latex-bold'>(D)</span>\ 300\qquad<span class='latex-bold'>(E)</span>\ 1800
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AMC 8 2003 Problem 10

<spanclass=′latex−bold′>BakeSale</span><span class='latex-bold'>Bake Sale</span> Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies di ffer, as shown.
∘\circ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy]
∘\circ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy]
∘\circ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy]
∘\circ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy]
Each friend uses the same amount of dough, and Art makes exactly 12 cookies. How many cookies will be in one batch of Trisha's cookies?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 12<spanclass=′latex−bold′>(C)</span> 16<spanclass=′latex−bold′>(D)</span> 18<spanclass=′latex−bold′>(E)</span> 24 <span class='latex-bold'>(A)</span>\ 10\qquad<span class='latex-bold'>(B)</span>\ 12\qquad<span class='latex-bold'>(C)</span>\ 16\qquad<span class='latex-bold'>(D)</span>\ 18\qquad<span class='latex-bold'>(E)</span>\ 24
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AMC 8 2003 Problem 9

<spanclass=′latex−bold′>BakeSale</span><span class='latex-bold'>Bake Sale</span> Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies di ffer, as shown.
∘\circ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy]
∘\circ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy]
∘\circ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy]
∘\circ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy]
Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Art's cookies sell for 60 cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?
<spanclass=′latex−bold′>(A)</span> 18<spanclass=′latex−bold′>(B)</span> 25<spanclass=′latex−bold′>(C)</span> 40<spanclass=′latex−bold′>(D)</span> 75<spanclass=′latex−bold′>(E)</span> 90 <span class='latex-bold'>(A)</span>\ 18\qquad<span class='latex-bold'>(B)</span>\ 25\qquad<span class='latex-bold'>(C)</span>\ 40\qquad<span class='latex-bold'>(D)</span>\ 75\qquad<span class='latex-bold'>(E)</span>\ 90
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AMC 8 2003 Problem 8

<spanclass=′latex−bold′>BakeSale</span><span class='latex-bold'>Bake Sale</span> Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies di ffer, as shown.
∘\circ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy]
∘\circ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy]
∘\circ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy]
∘\circ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy]
Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Who gets the fewest cookies from one batch of cookie dough?
<spanclass=′latex−bold′>(A)</span> Art<spanclass=′latex−bold′>(B)</span> Roger<spanclass=′latex−bold′>(C)</span> Paul<spanclass=′latex−bold′>(D)</span> Trisha<spanclass=′latex−bold′>(E)</span> There is a tie for fewest. <span class='latex-bold'>(A)</span>\ \text{Art}\qquad<span class='latex-bold'>(B)</span>\ \text{Roger}\qquad<span class='latex-bold'>(C)</span>\ \text{Paul}\qquad<span class='latex-bold'>(D)</span>\ \text{Trisha}\qquad<span class='latex-bold'>(E)</span>\ \text{There is a tie for fewest.}
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AMC 8 2003 Problem 24

A ship travels from point A to point B along a semicircular path, centered at Island X. Then it travels along a straight path from B to C. Which of these graphs best shows the ship's distance from Island X as it moves along its course?
[asy]size(150); pair X=origin, A=(-5,0), B=(5,0), C=(0,5); draw(Arc(X, 5, 180, 360)^^B--C); dot(X); label("XX", X, NE); label("CC", C, N); label("BB", B, E); label("AA", A, W);[/asy]
<spanclass=′latex−bold′>(A)</span><span class='latex-bold'>(A)</span> [asy] defaultpen(fontsize(7)); size(80); draw((0,16)--origin--(16,0), linewidth(0.9)); label("distance traveled", (8,0), S); label(rotate(90)*"distance to X", (0,8), W); draw(Arc((4,10), 4, 0, 180)^^(8,10)--(16,12)); [/asy]
<spanclass=′latex−bold′>(B)</span><span class='latex-bold'>(B)</span> [asy] defaultpen(fontsize(7)); size(80); draw((0,16)--origin--(16,0), linewidth(0.9)); label("distance traveled", (8,0), S); label(rotate(90)*"distance to X", (0,8), W); draw(Arc((12,10), 4, 180, 360)^^(0,10)--(8,10)); [/asy]
<spanclass=′latex−bold′>(C)</span><span class='latex-bold'>(C)</span> [asy] defaultpen(fontsize(7)); size(80); draw((0,16)--origin--(16,0), linewidth(0.9)); label("distance traveled", (8,0), S); label(rotate(90)*"distance to X", (0,8), W); draw((0,8)--(10,10)--(16,8)); [/asy]
<spanclass=′latex−bold′>(D)</span><span class='latex-bold'>(D)</span> [asy] defaultpen(fontsize(7)); size(80); draw((0,16)--origin--(16,0), linewidth(0.9)); label("distance traveled", (8,0), S); label(rotate(90)*"distance to X", (0,8), W); draw(Arc((12,10), 4, 0, 180)^^(0,10)--(8,10)); [/asy]
<spanclass=′latex−bold′>(E)</span><span class='latex-bold'>(E)</span> [asy] defaultpen(fontsize(7)); size(80); draw((0,16)--origin--(16,0), linewidth(0.9)); label("distance traveled", (8,0), S); label(rotate(90)*"distance to X", (0,8), W); draw((0,6)--(6,6)--(16,10)); [/asy]
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AMC 8 2003 Problem 25

In the figure, the area of square WXYZ is 25cm225 \text{cm}^2. The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In ΔABC\Delta ABC, AB=ACAB = AC, and when ΔABC\Delta ABC is folded over side BC, point A coincides with O, the center of square WXYZ. What is the area of ΔABC\Delta ABC, in square centimeters?
[asy] defaultpen(fontsize(8)); size(225); pair Z=origin, W=(0,10), X=(10,10), Y=(10,0), O=(5,5), B=(-4,8), C=(-4,2), A=(-13,5); draw((-4,0)--Y--X--(-4,10)--cycle); draw((0,-2)--(0,12)--(-2,12)--(-2,8)--B--A--C--(-2,2)--(-2,-2)--cycle); dot(O); label("AA", A, NW); label("OO", O, NE); label("BB", B, SW); label("CC", C, NW); label("WW",W , NE); label("XX", X, N); label("YY", Y, N); label("ZZ", Z, SE); [/asy]
<spanclass=′latex−bold′>(A)</span> 154<spanclass=′latex−bold′>(B)</span> 214<spanclass=′latex−bold′>(C)</span> 274<spanclass=′latex−bold′>(D)</span> 212<spanclass=′latex−bold′>(E)</span> 272 <span class='latex-bold'>(A)</span>\ \frac{15}4\qquad<span class='latex-bold'>(B)</span>\ \frac{21}4\qquad<span class='latex-bold'>(C)</span>\ \frac{27}4\qquad<span class='latex-bold'>(D)</span>\ \frac{21}2\qquad<span class='latex-bold'>(E)</span>\ \frac{27}2
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AMC 8 2003 Problem 22

The following figures are composed of squares and circles. Which figure has a shaded region with largest area? [asy]/* AMC8 2003 #22 Problem */ size(3inch, 2inch); unitsize(1cm); pen outline = black+linewidth(1); filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle, mediumgrey, outline); filldraw(shift(3,0)*((0,0)--(2,0)--(2,2)--(0,2)--cycle), mediumgrey, outline); filldraw(Circle((7,1), 1), mediumgrey, black+linewidth(1)); filldraw(Circle((1,1), 1), white, outline); filldraw(Circle((3.5,.5), .5), white, outline); filldraw(Circle((4.5,.5), .5), white, outline); filldraw(Circle((3.5,1.5), .5), white, outline); filldraw(Circle((4.5,1.5), .5), white, outline); filldraw((6.3,.3)--(7.7,.3)--(7.7,1.7)--(6.3,1.7)--cycle, white, outline); label("A", (1, 2), N); label("B", (4, 2), N); label("C", (7, 2), N); draw((0,-.5)--(.5,-.5), BeginArrow); draw((1.5, -.5)--(2, -.5), EndArrow); label("2 cm", (1, -.5));
draw((3,-.5)--(3.5,-.5), BeginArrow); draw((4.5, -.5)--(5, -.5), EndArrow); label("2 cm", (4, -.5));
draw((6,-.5)--(6.5,-.5), BeginArrow); draw((7.5, -.5)--(8, -.5), EndArrow); label("2 cm", (7, -.5));
draw((6,1)--(6,-.5), linetype("4 4")); draw((8,1)--(8,-.5), linetype("4 4"));[/asy]
<spanclass=′latex−bold′>(A)</span> A only<spanclass=′latex−bold′>(B)</span> B only<spanclass=′latex−bold′>(C)</span> C only<spanclass=′latex−bold′>(D)</span> both A and B<spanclass=′latex−bold′>(E)</span> all are equal <span class='latex-bold'>(A)</span>\ \text{A only}\qquad<span class='latex-bold'>(B)</span>\ \text{B only}\qquad<span class='latex-bold'>(C)</span>\ \text{C only}\qquad<span class='latex-bold'>(D)</span>\ \text{both A and B}\qquad<span class='latex-bold'>(E)</span>\ \text{all are equal}
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AMC 8 2003 Problem 13

Fourteen white cubes are put together to form the fi gure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces?
[asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(3/4,8/15,7/15); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(0,0,1)*unitcube, white, thick(), nolight); draw(shift(2,0,1)*unitcube, white, thick(), nolight); draw(shift(0,1,0)*unitcube, white, thick(), nolight); draw(shift(2,1,0)*unitcube, white, thick(), nolight); draw(shift(0,2,0)*unitcube, white, thick(), nolight); draw(shift(2,2,0)*unitcube, white, thick(), nolight); draw(shift(0,3,0)*unitcube, white, thick(), nolight); draw(shift(0,3,1)*unitcube, white, thick(), nolight); draw(shift(1,3,0)*unitcube, white, thick(), nolight); draw(shift(2,3,0)*unitcube, white, thick(), nolight); draw(shift(2,3,1)*unitcube, white, thick(), nolight);[/asy]
<spanclass=′latex−bold′>(A)</span> 4<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 8<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 12 <span class='latex-bold'>(A)</span>\ 4\qquad<span class='latex-bold'>(B)</span>\ 6\qquad<span class='latex-bold'>(C)</span>\ 8\qquad<span class='latex-bold'>(D)</span>\ 10\qquad<span class='latex-bold'>(E)</span>\ 12
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AMC 8 2003 Problem 15

A fi gure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a fi gure with the front and side views shown?
[asy] defaultpen(linewidth(0.8)); path p=unitsquare; draw(p^^shift(0,1)*p^^shift(1,0)*p); draw(shift(4,0)*p^^shift(5,0)*p^^shift(5,1)*p); label("FRONT", (1,0), S); label("SIDE", (5,0), S);[/asy]
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 7 <span class='latex-bold'>(A)</span>\ 3\qquad<span class='latex-bold'>(B)</span>\ 4\qquad<span class='latex-bold'>(C)</span>\ 5\qquad<span class='latex-bold'>(D)</span>\ 6\qquad<span class='latex-bold'>(E)</span>\ 7
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AMC 8 2003 Problem 21

The area of trapezoid ABCD ABCD is 164cm2 164 \text{cm}^2. The altitude is 8cm 8 \text{cm}, AB AB is 10cm 10 \text{cm}, and CD CD is 17cm 17 \text{cm}. What is BC BC, in centimeters? [asy]/* AMC8 2003 #21 Problem */ size(4inch,2inch); draw((0,0)--(31,0)--(16,8)--(6,8)--cycle); draw((11,8)--(11,0), linetype("8 4")); draw((11,1)--(12,1)--(12,0)); label("AA", (0,0), SW); label("DD", (31,0), SE); label("BB", (6,8), NW); label("CC", (16,8), NE); label("10", (3,5), W); label("8", (11,4), E); label("17", (22.5,5), E);[/asy]
<spanclass=′latex−bold′>(A)</span> 9<spanclass=′latex−bold′>(B)</span> 10<spanclass=′latex−bold′>(C)</span> 12<spanclass=′latex−bold′>(D)</span> 15<spanclass=′latex−bold′>(E)</span> 20 <span class='latex-bold'>(A)</span>\ 9\qquad<span class='latex-bold'>(B)</span>\ 10\qquad<span class='latex-bold'>(C)</span>\ 12\qquad<span class='latex-bold'>(D)</span>\ 15\qquad<span class='latex-bold'>(E)</span>\ 20
1
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2003 AMC 8 Problem 1-A cube

Jamie counted the number of edges of a cube, Jimmy counted the number of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum?
<spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 16<spanclass=′latex−bold′>(C)</span> 20<spanclass=′latex−bold′>(D)</span> 22<spanclass=′latex−bold′>(E)</span> 26<span class='latex-bold'>(A)</span>\ 12 \qquad <span class='latex-bold'>(B)</span>\ 16 \qquad <span class='latex-bold'>(C)</span>\ 20\qquad <span class='latex-bold'>(D)</span>\ 22 \qquad <span class='latex-bold'>(E)</span>\ 26
18
1

AMC 8 2003 Problem 18

Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party? [asy]/* AMC8 2003 #18 Problem */ pair a=(102,256), b=(68,131), c=(162,101), d=(134,150); pair e=(269,105), f=(359,104), g=(303,12), h=(579,211); pair i=(534, 342), j=(442,432), k=(374,484), l=(278,501); pair m=(282,411), n=(147,451), o=(103,437), p=(31,373); pair q=(419,175), r=(462,209), s=(477,288), t=(443,358); pair oval=(282,303); draw(l--m--n--cycle); draw(p--oval); draw(o--oval); draw(b--d--oval); draw(c--d--e--oval); draw(e--f--g--h--i--j--oval); draw(k--oval); draw(q--oval); draw(s--oval); draw(r--s--t--oval); dot(a); dot(b); dot(c); dot(d); dot(e); dot(f); dot(g); dot(h); dot(i); dot(j); dot(k); dot(l); dot(m); dot(n); dot(o); dot(p); dot(q); dot(r); dot(s); dot(t); filldraw(yscale(.5)*Circle((282,606),80),white,black); label(scale(0.75)*"Sarah", oval);[/asy]
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 7 <span class='latex-bold'>(A)</span>\ 1\qquad<span class='latex-bold'>(B)</span>\ 4\qquad<span class='latex-bold'>(C)</span>\ 5\qquad<span class='latex-bold'>(D)</span>\ 6\qquad<span class='latex-bold'>(E)</span>\ 7
20
1
19
1
16
1

Possible Seating Arrangements

Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has 44 seats: 11 Driver seat, 11 front passenger seat, and 22 back passenger seat. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 6<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 24<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 6 \qquad <span class='latex-bold'>(D)</span>\ 12 \qquad <span class='latex-bold'>(E)</span>\ 24
12
1

Tossing dice

When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the 55 faces than can be seen is divisible by 66?
<spanclass=′latex−bold′>(A)</span> 1/3<spanclass=′latex−bold′>(B)</span> 1/2<spanclass=′latex−bold′>(C)</span> 2/3<spanclass=′latex−bold′>(D)</span> 5/6<spanclass=′latex−bold′>(E)</span> 1<span class='latex-bold'>(A)</span>\ 1/3 \qquad <span class='latex-bold'>(B)</span>\ 1/2 \qquad <span class='latex-bold'>(C)</span>\ 2/3 \qquad <span class='latex-bold'>(D)</span>\ 5/6 \qquad <span class='latex-bold'>(E)</span>\ 1
11
1

Shoe Discount

Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by 1010 percent. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost 4040 dollars on Thursday?
<spanclass=′latex−bold′>(A)</span> 36<spanclass=′latex−bold′>(B)</span> 39.60<spanclass=′latex−bold′>(C)</span> 40<spanclass=′latex−bold′>(D)</span> 40.40<spanclass=′latex−bold′>(E)</span> 44<span class='latex-bold'>(A)</span>\ 36 \qquad <span class='latex-bold'>(B)</span>\ 39.60 \qquad <span class='latex-bold'>(C)</span>\ 40 \qquad <span class='latex-bold'>(D)</span>\ 40.40 \qquad <span class='latex-bold'>(E)</span>\ 44
7
1

Difference in tests

Blake and Jenny each took four 100100 point tests. Blake averaged 7878 on the four tests. Jenny scored 1010 points higher than Blake on the first test, 1010 points lower on the second test, and 2020 points higher on both the third and fourth test. What is the difference between Blake's average on the four tests and Jenny's average on the four tests?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 15<spanclass=′latex−bold′>(C)</span> 20<spanclass=′latex−bold′>(D)</span> 25<spanclass=′latex−bold′>(E)</span> 40<span class='latex-bold'>(A)</span>\ 10 \qquad <span class='latex-bold'>(B)</span>\ 15 \qquad <span class='latex-bold'>(C)</span>\ 20 \qquad <span class='latex-bold'>(D)</span>\ 25 \qquad <span class='latex-bold'>(E)</span>\ 40
5
1
4
1

Counting Wheels

A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted 77 children and 1919 wheels. How many tricycles were there?
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 6<spanclass=′latex−bold′>(E)</span> 7<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 6 \qquad <span class='latex-bold'>(E)</span>\ 7